3.2.9 Entropy fix

• For most Riemann problems (Fig. 26) the CIR scheme (185) actually produces initially the exact result.
• Only for a transonic rarefaction wave (as in the last example in Fig. 30) it differs.
• Godunov's idea: Solve a Riemann problem ($\Rightarrow$ Riemann solver) at every cell boundary and derive the corresponding flux over the boundary.
• For Burgers' equation the flux through the stagnation point ($v_{}=0$) in a transonic rarefaction wave is $f = \frac{1}{2} v_{}^2 = 0$.

$\Rightarrow$ Extend the CIR scheme to get a Godunov-type scheme by defining

$$f_{i+\frac{1}{2}}^n = \left\{ \begin{array}{ll} \frac{1}... ...\enspace v_{i} < 0 < v_{i+1}} \end{array} \right. \enspace .$$ (189)

The additional branch is an entropy fix to the CIR scheme.